Dynamical behavior of a mechanical system including Saint-Venant component coupled to a non-linear energy sink

• We study energy pumping between a non-linear system and a cubic NES.
• The main system includes a Saint-Venant element.
• Different invariant manifolds of the system at different time scales are revealed.
• We detect fixed points and fold singularities of the system.
• Periodic and/or strongly modulated regimes are traced.

Multi-scale vibratory energy exchange between a main oscillator including Saint-Venant term and a cubic non-linear energy sink is studied. Analytically obtained invariant manifold of the system at a fast time scale and detected fixed points and/or fold singularities at a first slow time scale let us predict and explain different regimes that the system may face during the energy exchange process. The paper will be accompanied by some numerical results confirming our analytical predictions.